1.4 Amdahl's law and Gustafson's law

Amdahl's law and Gustafson's law are key concepts in parallel computing that help us understand the limits and potential of speeding up programs using multiple processors. These laws provide insights into how much faster we can make our code run by adding more processing power.

Amdahl's law focuses on fixed-size problems, showing that speedup is limited by the sequential part of the code. Gustafson's law, on the other hand, considers scaled problems, suggesting that we can achieve better speedup by increasing both the problem size and number of processors together.

Amdahl's law

• Amdahl's law is a fundamental concept in parallel computing that describes the theoretical speedup of a program when utilizing multiple processors
• It provides insights into the limitations of parallelization and helps determine the maximum speedup that can be achieved by adding more processors to a system
• Amdahl's law is crucial for understanding the performance characteristics of parallel algorithms and the design considerations for exascale computing systems

Speedup limitations

Top images from around the web for Speedup limitations

• 

  Lei de Amdahl – Wikipédia, a enciclopédia livre View original

  Is this image relevant?

• 

  Amdahl's Law | Introduction to High Performance and High Throughput Computing View original

  Is this image relevant?

• 

  Amdahl's law - Wikipedia View original

  Is this image relevant?

• 

  Lei de Amdahl – Wikipédia, a enciclopédia livre View original

  Is this image relevant?

• 

  Amdahl's Law | Introduction to High Performance and High Throughput Computing View original

  Is this image relevant?

1 of 3

Top images from around the web for Speedup limitations

• 

  Lei de Amdahl – Wikipédia, a enciclopédia livre View original

  Is this image relevant?

• 

  Amdahl's Law | Introduction to High Performance and High Throughput Computing View original

  Is this image relevant?

• 

  Amdahl's law - Wikipedia View original

  Is this image relevant?

• 

  Lei de Amdahl – Wikipédia, a enciclopédia livre View original

  Is this image relevant?

• 

  Amdahl's Law | Introduction to High Performance and High Throughput Computing View original

  Is this image relevant?

1 of 3

• Amdahl's law states that the speedup of a program is limited by the sequential portion of the code that cannot be parallelized
• The sequential portion acts as a bottleneck and limits the overall speedup, regardless of the number of processors added
• The speedup is calculated as: $Speedup = \frac{1}{(1-P) + \frac{P}{N}}$, where $P$ is the fraction of parallelizable code and $N$ is the number of processors
• As the number of processors increases, the speedup approaches a theoretical limit determined by the sequential portion

Sequential vs parallel execution

• Amdahl's law distinguishes between the sequential and parallel portions of a program
• The sequential portion represents the code that must be executed sequentially and cannot be divided among multiple processors
• The parallel portion represents the code that can be executed concurrently by multiple processors
• The ratio of sequential to parallel code determines the potential speedup and the efficiency of parallelization

Fraction of parallelizable code

• The fraction of parallelizable code, denoted as $P$ in Amdahl's law, represents the portion of the program that can be executed in parallel
• A higher value of $P$ indicates a larger portion of the code that can be parallelized, leading to better speedup
• For example, if 90% of the code can be parallelized ($P = 0.9$), the theoretical speedup with an infinite number of processors is limited to 10 times
• Identifying and maximizing the parallelizable fraction is crucial for achieving optimal speedup in parallel computing

Asymptotic behavior

• Amdahl's law exhibits an asymptotic behavior, meaning that the speedup approaches a theoretical limit as the number of processors increases
• The sequential portion of the code becomes the dominant factor in determining the speedup, regardless of the number of processors
• For example, if 10% of the code is sequential ($P = 0.9$), the maximum speedup achievable is limited to 10 times, even with an infinite number of processors
• The asymptotic behavior highlights the diminishing returns of adding more processors beyond a certain point

Assumptions and limitations

• Amdahl's law makes several assumptions that may not always hold in practice
• It assumes that the problem size remains fixed and does not scale with the number of processors
• It assumes that the parallel portions of the code can be perfectly distributed among the processors without any overhead
• In reality, factors such as communication overhead, load imbalance, and synchronization costs can impact the actual speedup achieved
• Amdahl's law provides an upper bound on the speedup but does not consider the specific characteristics of the parallel algorithm or the underlying hardware

Gustafson's law

• Gustafson's law, also known as Gustafson-Barsis' law, is another important concept in parallel computing that addresses the limitations of Amdahl's law
• It focuses on the scaled speedup achieved when the problem size is increased along with the number of processors
• Gustafson's law is particularly relevant for exascale computing, where the goal is to solve larger and more complex problems using massive parallelism

Scaled speedup

• Gustafson's law introduces the concept of scaled speedup, which measures the speedup achieved when the problem size is increased proportionally to the number of processors
• The scaled speedup is calculated as: $Scaled Speedup = N - (1 - P)(N - 1)$, where $N$ is the number of processors and $P$ is the fraction of parallelizable code
• Unlike Amdahl's law, Gustafson's law suggests that the speedup can continue to increase as the problem size and the number of processors are scaled together

Problem size vs processor count

• Gustafson's law emphasizes the relationship between the problem size and the number of processors
• As the number of processors increases, the problem size is also increased to maintain a constant level of parallelism
• This allows the parallel portion of the code to scale with the number of processors, leading to improved speedup
• For example, doubling the number of processors while doubling the problem size can result in a nearly linear speedup

Parallel vs sequential workload

• Gustafson's law considers the ratio of parallel to sequential workload in a program
• The parallel workload represents the portion of the code that can be executed concurrently by multiple processors
• The sequential workload represents the portion of the code that must be executed sequentially, regardless of the number of processors
• Gustafson's law assumes that the sequential workload remains constant while the parallel workload increases with the problem size

Assumptions and implications

• Gustafson's law makes assumptions about the scalability of the problem and the parallel algorithm
• It assumes that the problem size can be increased to maintain a constant level of parallelism
• It assumes that the parallel algorithm can efficiently utilize the additional processors and problem size
• Gustafson's law implies that parallel computing can achieve significant speedup for large-scale problems that can be divided into smaller, independent tasks
• It highlights the importance of designing algorithms and systems that can scale with the problem size and the number of processors

Comparison of laws

• Amdahl's law and Gustafson's law provide different perspectives on the speedup achievable through parallelization
• They have distinct focuses and make different assumptions about the problem size and the parallel workload
• Understanding the differences between these laws is crucial for designing and analyzing parallel algorithms and systems

Speedup focus: fixed vs scaled

• Amdahl's law focuses on the speedup achieved with a fixed problem size, where the workload remains constant as the number of processors increases
• Gustafson's law focuses on the scaled speedup achieved when the problem size is increased along with the number of processors
• Amdahl's law emphasizes the limitations imposed by the sequential portion of the code, while Gustafson's law emphasizes the potential for speedup through problem scaling

Parallelization assumptions

• Amdahl's law assumes that the problem size remains fixed and that the parallel portions of the code can be perfectly distributed among the processors
• Gustafson's law assumes that the problem size can be increased to maintain a constant level of parallelism and that the parallel algorithm can efficiently utilize the additional processors
• These assumptions impact the predicted speedup and the applicability of the laws to different scenarios

Applicability to different scenarios

• Amdahl's law is more applicable to scenarios where the problem size is fixed and cannot be easily scaled (fixed-size problems)
• Gustafson's law is more applicable to scenarios where the problem size can be increased to take advantage of additional processors (scaled-size problems)
• The choice of which law to apply depends on the specific characteristics of the problem and the parallel algorithm

Limitations and criticisms

• Both Amdahl's law and Gustafson's law have limitations and have been subject to criticisms
• They make simplifying assumptions about the parallel workload distribution and the overhead associated with parallelization
• They do not consider factors such as communication overhead, load imbalance, and synchronization costs, which can impact the actual speedup achieved
• The laws provide theoretical upper bounds on speedup but may not accurately predict the performance of real-world parallel systems

Impact on parallel computing

• Amdahl's law and Gustafson's law have had a significant impact on the field of parallel computing
• They provide guiding principles and insights for the design and analysis of parallel algorithms and systems
• Understanding these laws is essential for making informed decisions about parallelization strategies and resource allocation in exascale computing

Guiding principles for parallelization

• Amdahl's law emphasizes the importance of identifying and minimizing the sequential portion of the code to maximize speedup
• Gustafson's law highlights the potential for achieving speedup by scaling the problem size along with the number of processors
• These principles guide the development of parallel algorithms and the allocation of computational resources
• They encourage the identification of parallelizable tasks and the design of algorithms that can scale with the problem size

Influence on algorithm design

• The insights provided by Amdahl's law and Gustafson's law influence the design of parallel algorithms
• Algorithms are designed to maximize the parallel portion of the code and minimize the sequential bottlenecks
• Techniques such as data partitioning, load balancing, and communication optimization are employed to improve parallel efficiency
• The laws guide the selection of appropriate parallelization strategies based on the characteristics of the problem and the available resources

Considerations for exascale systems

• Exascale computing systems, which aim to achieve a billion billion (10^18) calculations per second, pose unique challenges for parallel computing
• Amdahl's law and Gustafson's law provide insights into the scalability and performance considerations for exascale systems
• The laws highlight the need for algorithms and software that can effectively utilize the massive parallelism offered by exascale systems
• Factors such as communication overhead, data movement, and energy efficiency become critical at the exascale level
• The laws guide the design of exascale algorithms and the optimization of parallel performance

Balancing parallelism and overhead

• Amdahl's law and Gustafson's law emphasize the trade-offs between parallelism and overhead in parallel computing
• Increasing the level of parallelism can lead to diminishing returns due to the overhead associated with communication, synchronization, and load imbalance
• The laws provide insights into finding the optimal balance between parallelism and overhead
• Techniques such as granularity control, communication optimization, and load balancing are employed to minimize overhead and maximize parallel efficiency
• The laws guide the selection of appropriate parallelization granularity and the management of parallel resources

Real-world applications

• Amdahl's law and Gustafson's law have practical implications for real-world parallel computing applications
• They are used to analyze and optimize the performance of parallel algorithms and systems in various domains
• Case studies and examples demonstrate the application of these laws in practice

Case studies demonstrating laws

• Scientific simulations: Parallel simulations of complex physical phenomena, such as climate modeling or molecular dynamics, often involve large problem sizes that can be scaled with the number of processors. Gustafson's law is relevant in these scenarios
• Data analytics: Big data processing and analytics tasks, such as machine learning or graph analysis, can benefit from parallelization. Amdahl's law helps identify the sequential bottlenecks and guides the optimization of parallel algorithms
• Computer graphics: Rendering complex 3D scenes or performing real-time image processing can leverage parallel computing. Amdahl's law and Gustafson's law guide the parallelization strategies and the distribution of workload among multiple processors

Optimization strategies based on laws

• Identifying sequential bottlenecks: Amdahl's law emphasizes the importance of identifying and minimizing the sequential portion of the code. Profiling tools and performance analysis techniques are used to identify and optimize sequential bottlenecks
• Problem decomposition: Gustafson's law suggests that problem decomposition and scaling the problem size can lead to improved speedup. Techniques such as domain decomposition, data partitioning, and task parallelism are employed to decompose the problem and distribute the workload among processors
• Load balancing: Both laws highlight the impact of load imbalance on parallel performance. Load balancing techniques, such as dynamic load balancing or work stealing, are used to ensure an even distribution of workload among processors and minimize idle time

Challenges in applying laws

• Complexity of real-world applications: Real-world applications often have complex dependencies, data structures, and communication patterns that make it challenging to apply Amdahl's law and Gustafson's law directly
• Non-uniform memory access (NUMA): In large-scale parallel systems, memory access latencies can vary depending on the location of the data relative to the processor. NUMA effects can impact the parallel performance and require careful data placement and memory management strategies
• Scalability limitations: As the number of processors increases, factors such as communication overhead, synchronization, and resource contention can limit the scalability of parallel algorithms. The laws provide upper bounds on speedup, but achieving those bounds in practice requires careful optimization and tuning

Future directions and research areas

• Heterogeneous computing: The integration of different types of processors, such as CPUs, GPUs, and accelerators, presents new challenges and opportunities for parallel computing. Extending Amdahl's law and Gustafson's law to heterogeneous environments is an active area of research
• Energy efficiency: Energy consumption is a critical concern in exascale computing. Researching energy-efficient parallel algorithms and power management techniques based on the insights from Amdahl's law and Gustafson's law is crucial for sustainable exascale computing
• Scalable algorithms: Developing parallel algorithms that can scale to exascale levels requires addressing challenges such as communication efficiency, data locality, and fault tolerance. Amdahl's law and Gustafson's law provide guidance for designing scalable algorithms and optimizing their performance
• Machine learning and AI: The growing demand for machine learning and artificial intelligence workloads presents new opportunities for parallel computing. Applying the principles of Amdahl's law and Gustafson's law to the parallelization of machine learning algorithms and the optimization of AI frameworks is an active research area